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euclidean distance in feature space

hi everyone,

i got this equation from the article " Nonlinear Dimensionality Reduction for Classification Using Kernel Weighted Subspace Method " to solve euclidean distance between the mean of class i and j in feature space. Each of class i and j consist of several column vectors. Here is the equation :

Attachment 9716

where Ni and Nj = the number of column vectors in class i and j, and x is a column vector.

The above equation is solved using "kernel trick" by substituting k(xi,xj) with the appropriate kernel. In the article itself, the author is using RBF kernel :$\displaystyle k(xi, xj) = exp( ||xi - xj||^2 / 10^9) $

Given this example :

Class 1 consist of

x1 = { 3, 4, 1 } (column vector 1 in class 1)

x2 = { 9, 2, 3 } (column vector 2 in class 1)

Class 2 consist of

x1 = { 2, 6, 5 } (column vector 1 in class 2)

x2 = { 7, 1, 3 } (column vector 2 in class 2)

I wonder how to obtain the euclidean distance d(i,j) above?

I try this following approach :

a) sum each element of column vector in each class

sum of class 1 (denoted as s1) = { 12, 6, 4 }

sum of class 2 (s2) = { 9, 7, 8 }

b) apply the obtained sum to the equation, so that the sum of k[i1,i2] = k[s1,s1]; the sum of k[j1,j2] = k[s2,s2]; the sum of k[i1,j1] = k[s1,s2]; the sum of k[j1,i1] = k[s2,s1].

But here come the problem, let's say that k[s1,s1] = u, k[s2,s2] = v, k[s1,s2] = x, and k[s2,s1] = y.

Applying RBF kernel into those equation, I get this resulting value :

u = $\displaystyle exp (0)$ = 1;

v = $\displaystyle exp (0)$ = 1;

x = $\displaystyle exp(26 / 10^9) = 1.000000026$;

and y = $\displaystyle exp(26 / 10^9) = 1.000000026$;

So d(i,j) = $\displaystyle sqrt( (1/Ni^2) * u + (1/Nj^2) * v - (1/(Ni*Nj)) * x - (1/(Ni*Nj)) * y )$

= $\displaystyle sqrt( 1/4 + 1/4 - (1/4) * 1.000000026 - (1/4) * 1.000000026)$

= $\displaystyle sqrt( -0.000000014 )$

So, I got stuck in this square root of negative value result. I would appreciate any help on this matter.

Thanks in advance!

(Sorry for my poor English)